Technically, there is no difference between “up” and “straight up” when making a cocktail. Asking for a drink served “up” or “straight up” simply means that you want the drink shaken or stirred over ice, and then strained and served sans ice in a cocktail glass. Classic cocktails that are almost always served up include the Martini and the Daiquiri. Others that are dependent on personal preference include drinks such as the Negroni and the Manhattan.
Where the terminology “up” and “straight up” starts to differ, however, is with the word “straight.” The term “straight,” when it comes to drinks, means a person would like a spirit served solo, without any mixer. Asking for a whiskey “straight up,” means that the consumer would like their whiskey to be chilled first, then served to them in the appropriate glassware.
Where one can run into trouble using the terms “up” and “straight up” interchangeably is when it comes to cocktails like the Martini. If one were to order a vodka or gin Martini “straight up” it’s very likely the bartender could interpret this order to mean the drinker would like vodka or gin chilled and then served to them in a Martini glass, no vermouth. This might be the preferred vodka Martini for many people, but if it isn’t for you, ask for the cocktail to be served “up” instead.